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Symmetry is a central concept for classical and quantum field theory,usually, symmetry is described by a finite group or Lie group. In this work, weintroduce the weak Hopf algebra extension of symmetry, which arises naturallyin anyonic quantum systems; and we establish weak Hopf symmetry breaking theorybased on the fusion closed set of anyons. As a concrete example, we implement athorough investigation of the quantum double model based on a given weak Hopfalgebra and show that the vacuum sector of the model has weak Hopf symmetry.The topological excitations and ribbon operators are discussed in detail. Thegapped boundary and domain wall theories are also established, we show that thegapped boundary is algebraically determined by a comodule algebra, orequivalently, a module algebra; and the gapped domain wall is determined by thebicomodule algebra, or equivalently, a bimodule algebra. The microscopiclattice constructions of the gapped boundary and domain wall are discussed indetail. We also introduce the weak Hopf tensor network states, via which wesolve the weak Hopf quantum double lattice models on closed and open surfaces.The duality of the quantum double phases is discussed in the last part.

On weak Hopf symmetry and weak Hopf quantum double model